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Let us say we are given a sequence of order 2. By order 2 I mean that we have a sequence in which the differences between the terms forms a sequence of order 1, which has a constant difference between terms. How can I prove that the nth term of a sequence of order 2 can be represented as:

an^2 + bn + c?

Or more generally how would I prove that that the nth term of a sequence of order k can be represented as:

an^k + bn^k-1 +.... + pn + q?

Any help would we greatly appreciated.

Thanks